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330                        ANALYTICAL MECHANICS
Let m be the mass of the vibrating cylinder and a and h the radii of vibrating and the fixed cylinders, respectively.   Then at any instant
T = 1 /co2,
where T denotes the kinetic energy of the vibrating cylinder, / its moment of inertia about the element of contact and w its angular velocity. But
and                  co = - = (b — «)0,                          ™T/,  •< <n
(I      ^           '   '                                 r IG.  14o.
where v is the linear velocity of the axis of the moving cylinder and i angular velocity.   Therefore
On the other hand we have the following expressions for the* poten
-energy :
U = nigh
~ mg (b — «} (1 — oos0)
•'Since 0 is supposed to remain small all the time, it is permissible to neg the higher terms of 6 in the last expression for (/.   Then^fore wc^ hav
U = i mg (h - a) 0\
'Thus both T and U are expressed in forms which are adapted to the ap •cation of Lagrange's method.
The total energy of the system is
E = f m (b - a)2^2 + | mg (b - «) 0\ Differentiating the last equation with respect to the time we obtain
3(/>-«)0 + 200« 0. Therefore
* The expansion is (tarried out by Maehuirin'n Theorem.  Ke<» Appendix